Improved Newton Iterative Algorithm for Fractal Art Graphic Design

Chen, Huijuan; Zheng, Xinqian

doi:10.1155/2020/6623049

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…Such 2D regions are esthetically so beautiful that their applications are not only found in applied mathematics, but also people working in fields such as architecture, arts, and design also use the concept to obtain pleasing designs. Many other fields of applications for these basins can be seen in [31][32][33][34][35][36] and most of the references cited therein. It may be noted that linear models do not depict such dynamically eye-catching behavior, whereas the non-linearity results in features such as those seen herein under the proposed iterative method PM 10 .…”

Section: Basins Of Attractionmentioning

confidence: 99%

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

et al. 2021

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There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models.

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Section: Basins Of Attractionmentioning

confidence: 99%

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

et al. 2021

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“…This concept is found to have several applications in scientific fields including applied mathematics, arts, design, architecture, and many more. Interesting applications of basins of attraction can be seen in the references [25][26][27][28][29] and most of the references cited therein. Nonlinear models under both scalar and vector cases have aesthetically interesting behavior that is not usually observed in the linear models.…”

Section: Nr2mentioning

confidence: 99%

A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

2021

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Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology.

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Utilizing Fractal Dimensions as Indicators to Detect Elements of Visual Attraction: A Case Study of the Greenway along Lake Taihu, China

Ren

Yocom

Guo

2023

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It is widely acknowledged that the quality of greenway landscape resources enhances the visual appeal of people. While most studies have evaluated visual perception and preference, few have considered the relationship between the distribution of greenways in relation to the proximity of water bodies such as lakes and rivers. Such an investigation requires an in-depth analysis of how to plan and design greenways in order to better enhance people’s willingness to access and utilize them. In this research we propose specific color brightness and contour visual attraction elements to further discuss the quality of greenway landscape resources in the rapidly urbanizing Lake Taihu region of China. Specifically, we utilize a common method in fractal theory analysis called counting box dimension to calculate and analyze the sample images. The method generates data on fractal dimension (FD) values of two elements; the optimal fractal dimension threshold range; the characteristics exhibited by the maximum and minimum fractal dimension values in the greenway landscape; and the relationship between the two visual attraction elements allowing us to derive distribution of the greenway and water bodies. The results reveal that greenway segments with high values of the visual attraction element of color brightness fractal dimension (FD) are significantly closer to the lake than those subject to high values of the visual attraction element. Some segments are even close to the lake surface, which is because the glare from the direct sunlight and the reflection from the lake surface superimposed on each other, so that the greenway near the lake surface is also affected by the brightness and shows the result of high color brightness values. However, the greenway segments with high values of contour element FD are clearly more influenced by plants and other landscape elements. This is due to the rich self-similarity of the plants themselves. Most of the greenway segments dominated by contour elements are distant from the lake surface. Both color brightness and contour elements are important indicators of the quality of the visual resources of the Lake Taihu Greenway landscape. This reveals that the determination of the sub-dimensional values of color brightness (1.7608, 1.9337) and contour (1.7230, 1.9006) visual attraction elements and the optimal threshold range (1.7608, 1.9006) can provide theoretical implications for the landscape planning and design of lake-ring type greenways and practical implications for assessing the quality of visual resources in greenway landscapes.

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Improved Newton Iterative Algorithm for Fractal Art Graphic Design

Cited by 3 publications

References 19 publications

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

Utilizing Fractal Dimensions as Indicators to Detect Elements of Visual Attraction: A Case Study of the Greenway along Lake Taihu, China

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